2.3.2. Coupling Coordination Degree Model of the PLEF

The concept of coupling is derived from physics and refers to the phenomenon where two or more systems interact and influence each other. The degree of coupling is used to measure the degree of interaction and mutual influence between multiple systems [51]. However, it cannot reflect the synergic relationship between multiple systems [28]. Therefore, we introduced the coupling coordination index to construct a coupling coordination model. First, by evaluating the production, living and ecological function values of each geographic grid, the coupling coordination model is used to measure the degree of coupling and coordination among the PLEF of each geographic grid to reveal the degree of coordinated development of the PLEF. Second, the coupling coordination degree was divided into a coordination zone, break-in zone and incongruous zone by using the natural breakpoint classification method. Coordination zone refers to the coupling among the PLEF is becoming stronger and developing in an orderly direction, and it is in a period of high-level coupling and coordination. Break-in zone means that the PLEF began to develop in coordination and shows the characteristics of benign coupling and coordination. Incongruous zone refers to the interaction of the PLEF is low, and it is in a state of disorderly development. The specific calculation process is shown in Formulas (2) to (4).

$$\mathcal{C} = 3 \left\{ \frac{P\_{\bar{i}} \cdot R\_{\bar{i}} \cdot E\_{\bar{i}}}{\left(P\_{\bar{i}} + R\_{\bar{i}} + E\_{\bar{i}}\right)^{3}} \right\}^{1/3} \tag{2}$$

$$T = \alpha P\_i + \beta R\_i + \gamma E\_i \tag{3}$$

$$D = \left(\mathbb{C} \cdot \mathbb{T}\right)^{1/2} \tag{4}$$

In Formula (2), *C* represents the degree of coupling, with *C* ∈ [0, 1]; the larger the value of *C*, the stronger the interaction among the PLEF. *Pi*, *Ri* and *Ei* represent the evaluation scores of productive, living and ecological functions, respectively. In Formula (3), T represents the coordination index among the PLEF. *α*, *β* and *γ* respectively refer to the undetermined coefficients of the production function, living function and ecological function. Due to the lack of unified standards and methods for determining the undetermined coefficients, Wang Cheng et al. [22,28] determined the undetermined coefficients based on the opinions of experienced experts, the main functions of the research area and the contribution of PLEF, and their method is certainly reasonable. Therefore, based on the existing research results [22,28] and experts asked for advice, the undetermined coefficients were determined to be α = β = 0.3 and γ = 0.4. In Formula (4), *D* refers to the degree of coupling and coordination among the PLEF.

Moreover, it is of great significance to further analyze the strengths and weaknesses of the coupling coordination status between every two of the PLEF [28]. Therefore, to explore the coupling coordination status between every two of the PLEF, the coupling coordination degree model is further evolved, as shown in Formulas (5) to (6).

$$\mathbf{C}\_{1} = 2 \left\{ \frac{P\_{i} \cdot R\_{i}}{\left(P\_{i} + R\_{i}\right)^{2}} \right\}^{1/2} \mathbf{C}\_{2} = 2 \left\{ \frac{R\_{i} \cdot E\_{i}}{\left(R\_{i} + E\_{i}\right)^{2}} \right\}^{1/2} \mathbf{C}\_{3} = 2 \left\{ \frac{P\_{i} \cdot E\_{i}}{\left(P\_{i} + E\_{i}\right)^{2}} \right\}^{1/2} \tag{5}$$

$$D = \left(\mathbb{C} \cdot \mathbb{T}\right)^{1/2}, \ T\_1 = aP\_i + \beta R\_i \text{ or } T\_2 = aP\_i + \gamma E\_i \text{ or } T\_3 = \beta R\_i + \gamma E\_i \tag{6}$$

Based on the existing research results [19,26], when measuring the degree of coupling and coordination between the production function and living function, α = β = 0.5. When measuring the degree of coupling and coordination between the production function and ecological function, α = 0.45 and γ = 0.55. When measuring the degree of coupling and coordination between the living function and ecological function, β = 0.45 and γ = 0.55.
